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Generalized Linear Models - 1 (Parameter Estimation (Need to use Maximum…
Generalized Linear Models - 1
Generalising the GLM
Looking Beyond
continuous outcomes
Although indiv outcome
is Y/N, often use proportion
as the outcome
In order to understand population:
Proporation
(mortality) rate
Disease ?Y/N
Health studies
look for specific outcome
g(µ) = Xβ
g() is a function that
links
E(Y) = µ to X in a linear fashion
can easily change g to
other (non-linear) functions
g() is
link function
g() must be monotonic & differentiable
GLM
Use
Principle of least squares
to estimate values for β
Fitted model gives
line of best fit
determined by minimising
the error sums of squares (SSE)
Fitted values for β are called
least squares estimates
Link Function
Binary outcome va - Binomial Dist.
Logit link function
Logistic regression
Transformation:
Each type of data
has a distribution &
a natural link function
(commonly used but
not absolute requirement)
eg if outcome va
is a probability, never >1,
non-linear relationship
Transform Y using
logit function
relationship is
now straight line
then use linear model
Parameter Estimation
Need to use
Maximum Likelihood Estimation
(MLE)
MLE - what is the most likely value
of the βs given the observed data?
Cannot use Least Squares Estimation
to estimate the values of β
To do the MLE:
link function must be
monotonic & differentiable
Distribution must belong to
exponential family
Summary
Type of link function
changes for different
types of data
Provides options for
modelling more types of
outcome data
Differ from GLM
Use a link function
(transforming Y)
Use MLE instead of
least squares estimation
Downside
Some model-specific
pseudo R-squared values
May need to use
deviance
to
measure model fit
Cannot use R-squared
to measure explanatory power